Template:Goodness-of-fit test camsaa

Goodness-of-Fit Tests

While using the Crow-AMSAA model in the RGA 7 software, there are four goodness-of-fit tests which may become available depending on their applicability. The Cramér-von Mises goodness-of-fit test tests the hypothesis that the data follows a nonhomogeneous Poisson process with failure intensity equal to [math]\displaystyle{ u(t)=\lambda \beta {{t}^{\beta -1}} }[/math] . This test can be applied when the failure data is complete over the continuous interval [math]\displaystyle{ [0,{{T}_{q}}] }[/math] with no gaps in the data. The Chi-Squared test is a goodness-of-fit test that can be applied under more general circumstances, particularly when the data set is grouped. In addition, for multiple system data the Common Beta Hypothesis (CBH) test also can be used to compare the intensity functions of the individual systems by comparing the [math]\displaystyle{ {{\beta }_{q}} }[/math] results for each system. Lastly, the Laplace Trend test checks for trends within the data. Due to their general application to multiple models, the Common Beta Hypothesis test and the Laplace Trend test are both presented in Appendix B. The Cramér-von Mises and Chi-Squared tests are described here since they apply to the Crow-AMSAA model only.

Cramér-von Mises Test for Individual Failure Times

If the individual failure times are known, a Cramér-von Mises statistic is used to test the null hypothesis that a non-homogeneous Poisson process with failure intensity function [math]\displaystyle{ \rho \left( t \right)=\lambda \,\beta \,{{t}^{\beta -1}}\left( \lambda \gt 0,\beta \gt 0,t\gt 0 \right) }[/math] properly describes the reliability growth of a system. The Cramér-von Mises goodness-of-fit statistic is then given by the following expression:

[math]\displaystyle{ C_{M}^{2}=\frac{1}{12M}+\underset{i=1}{\overset{M}{\mathop \sum }}\,{{\left[ {{\left( \frac{{{T}_{i}}}{T} \right)}^{{\hat{\beta }}}}-\frac{2i-1}{2M} \right]}^{2}} }[/math]

where:

[math]\displaystyle{ M=\left\{ \begin{matrix} N\text{ if the test is time terminated} \\ N-1\text{ if the test is failure terminated} \\ \end{matrix} \right\} }[/math]

The failure times, [math]\displaystyle{ {{T}_{i}} }[/math] , must be ordered so that [math]\displaystyle{ {{T}_{1}}\lt {{T}_{2}}\lt \ldots \lt {{T}_{M}} }[/math] . If the statistic [math]\displaystyle{ C_{M}^{2} }[/math] exceeds the critical value corresponding to [math]\displaystyle{ M }[/math] for a chosen significance level, then the null hypothesis that the Crow-AMSAA model adequately fits the data shall be rejected. Otherwise, the model shall be accepted. Critical values of [math]\displaystyle{ C_{M}^{2} }[/math] are shown in Appendix B, Table B.2, where the table is indexed by the total number of observed failures, [math]\displaystyle{ M }[/math] .

Chi-Squared Test for Grouped Data

A Chi-Squared goodness-of-fit test is used to test the null hypothesis that the Crow-AMSAA reliability model adequately represents a set of grouped data. The expected number of failures in the interval from [math]\displaystyle{ {{T}_{i-1}} }[/math] to [math]\displaystyle{ {{T}_{i}} }[/math] is approximated by:

[math]\displaystyle{ {{\widehat{\theta }}_{i}}=\hat{\lambda }\left( T_{i}^{{\hat{\beta }}}-T_{i-1}^{{\hat{\beta }}} \right) }[/math]

For each interval, [math]\displaystyle{ {{\widehat{\theta }}_{i}} }[/math] shall not be less than 5 and, if necessary, adjacent intervals may have to be combined so that the expected number of failures in any combined interval is at least 5. Let the number of intervals after this recombination be [math]\displaystyle{ d }[/math] and let the observed number of failures in the [math]\displaystyle{ {{i}^{th}} }[/math] new interval be [math]\displaystyle{ {{N}_{i}} }[/math] and let the expected number of failures in the [math]\displaystyle{ {{i}^{th}} }[/math] new interval be [math]\displaystyle{ {{\widehat{\theta }}_{i}} }[/math] . Then the following statistic is approximately distributed as a Chi-Squared random variable with degrees of freedom [math]\displaystyle{ d-2 }[/math] .

[math]\displaystyle{ {{\chi }^{2}}=\underset{i=1}{\overset{d}{\mathop \sum }}\,\frac{{{({{N}_{i}}-{{\widehat{\theta }}_{i}})}^{2}}}{{{\widehat{\theta }}_{i}}} }[/math]

The null hypothesis is rejected if the [math]\displaystyle{ {{\chi }^{2}} }[/math] statistic exceeds the critical value for a chosen significance level. This means that the hypothesis that the Crow-AMSAA model adequately fits the grouped data shall be rejected. Critical values for this statistic can be found in tables of the Chi-Squared distribution.

Example 6
An aircraft has scheduled inspections at intervals of 20 flight hours. Table 5.4 gives the data set from the first 100 hours of flight testing.

Table 5.4 - Grouped test data

Start Time	End Time	Number of Failures
0	20	13
20	40	16
40	60	5
60	80	8
80	100	7

1) Estimate the parameters of the Crow-AMSAA model using maximum likelihood estimation.

2) Evaluate the goodness-of-fit.

Solution

1) Obtain the estimator of [math]\displaystyle{ \beta }[/math] using Eqn. (vv). Using RGA the value of [math]\displaystyle{ \widehat{\beta } }[/math] is 0.75285. Now plug this value into Eqn. (vv1) and [math]\displaystyle{ \widehat{\lambda } }[/math] is:

[math]\displaystyle{ \widehat{\lambda }=1.52931 }[/math]

2) There are a total of observed failures from [math]\displaystyle{ d=5 }[/math] intervals. Table 5.5 shows that those adjacent intervals do not have to be combined after applying Eqn. (seta) to the original intervals.

Table 5.5 - Observed vs. Expected Number of Failures for Grouped data

Start Time	End Time	Observed Number of Failures	Expected Number of Failures
0	20	13	14.59
20	40	16	9.99
40	60	5	8.77
60	80	8	8.07
80	100	7	7.58

To test the model's goodness-of-fit, a Chi-Squared statistic of 5.45 is compared to the critical value of 7.8 corresponding to 3 degrees of freedom and a 0.05 significance level. Since the statistic is less than the critical value, the applicability of the Crow-AMSAA model is accepted.